3.3.47 \(\int \cosh (x) \text {csch}(4 x) \, dx\) [247]

Optimal. Leaf size=26 \[ -\frac {1}{4} \tanh ^{-1}(\cosh (x))+\frac {\tanh ^{-1}\left (\sqrt {2} \cosh (x)\right )}{2 \sqrt {2}} \]

[Out]

-1/4*arctanh(cosh(x))+1/4*arctanh(cosh(x)*2^(1/2))*2^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1107, 212} \begin {gather*} \frac {\tanh ^{-1}\left (\sqrt {2} \cosh (x)\right )}{2 \sqrt {2}}-\frac {1}{4} \tanh ^{-1}(\cosh (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cosh[x]*Csch[4*x],x]

[Out]

-1/4*ArcTanh[Cosh[x]] + ArcTanh[Sqrt[2]*Cosh[x]]/(2*Sqrt[2])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 1107

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/(b/
2 - q/2 + c*x^2), x], x] - Dist[c/q, Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*
a*c, 0] && PosQ[b^2 - 4*a*c]

Rubi steps

\begin {align*} \int \cosh (x) \text {csch}(4 x) \, dx &=-\text {Subst}\left (\int \frac {1}{-4+12 x^2-8 x^4} \, dx,x,\cosh (x)\right )\\ &=2 \text {Subst}\left (\int \frac {1}{4-8 x^2} \, dx,x,\cosh (x)\right )-2 \text {Subst}\left (\int \frac {1}{8-8 x^2} \, dx,x,\cosh (x)\right )\\ &=-\frac {1}{4} \tanh ^{-1}(\cosh (x))+\frac {\tanh ^{-1}\left (\sqrt {2} \cosh (x)\right )}{2 \sqrt {2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.16, size = 183, normalized size = 7.04 \begin {gather*} \frac {2 i \text {ArcTan}\left (\frac {\cosh \left (\frac {x}{2}\right )+\sinh \left (\frac {x}{2}\right )}{\left (1+\sqrt {2}\right ) \cosh \left (\frac {x}{2}\right )-\left (-1+\sqrt {2}\right ) \sinh \left (\frac {x}{2}\right )}\right )-2 i \text {ArcTan}\left (\frac {\cosh \left (\frac {x}{2}\right )+\sinh \left (\frac {x}{2}\right )}{\left (-1+\sqrt {2}\right ) \cosh \left (\frac {x}{2}\right )-\left (1+\sqrt {2}\right ) \sinh \left (\frac {x}{2}\right )}\right )+4 \tanh ^{-1}\left (\sqrt {2}-i \tanh \left (\frac {x}{2}\right )\right )-2 \sqrt {2} \log \left (\cosh \left (\frac {x}{2}\right )\right )-\log \left (\sqrt {2}-2 \cosh (x)\right )+\log \left (\sqrt {2}+2 \cosh (x)\right )+2 \sqrt {2} \log \left (\sinh \left (\frac {x}{2}\right )\right )}{8 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]*Csch[4*x],x]

[Out]

((2*I)*ArcTan[(Cosh[x/2] + Sinh[x/2])/((1 + Sqrt[2])*Cosh[x/2] - (-1 + Sqrt[2])*Sinh[x/2])] - (2*I)*ArcTan[(Co
sh[x/2] + Sinh[x/2])/((-1 + Sqrt[2])*Cosh[x/2] - (1 + Sqrt[2])*Sinh[x/2])] + 4*ArcTanh[Sqrt[2] - I*Tanh[x/2]]
- 2*Sqrt[2]*Log[Cosh[x/2]] - Log[Sqrt[2] - 2*Cosh[x]] + Log[Sqrt[2] + 2*Cosh[x]] + 2*Sqrt[2]*Log[Sinh[x/2]])/(
8*Sqrt[2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(52\) vs. \(2(18)=36\).
time = 0.85, size = 53, normalized size = 2.04

method	result	size
risch	\(\frac {\ln \left ({\mathrm e}^{x}-1\right )}{4}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{4}+\frac {\ln \left (1+{\mathrm e}^{2 x}+{\mathrm e}^{x} \sqrt {2}\right ) \sqrt {2}}{8}-\frac {\ln \left (1+{\mathrm e}^{2 x}-{\mathrm e}^{x} \sqrt {2}\right ) \sqrt {2}}{8}\)	\(53\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)*csch(4*x),x,method=_RETURNVERBOSE)

[Out]

1/4*ln(exp(x)-1)-1/4*ln(exp(x)+1)+1/8*ln(1+exp(2*x)+exp(x)*2^(1/2))*2^(1/2)-1/8*ln(1+exp(2*x)-exp(x)*2^(1/2))*
2^(1/2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (18) = 36\).
time = 0.46, size = 60, normalized size = 2.31 \begin {gather*} \frac {1}{8} \, \sqrt {2} \log \left (\sqrt {2} e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) - \frac {1}{8} \, \sqrt {2} \log \left (-\sqrt {2} e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) - \frac {1}{4} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac {1}{4} \, \log \left (e^{\left (-x\right )} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*csch(4*x),x, algorithm="maxima")

[Out]

1/8*sqrt(2)*log(sqrt(2)*e^(-x) + e^(-2*x) + 1) - 1/8*sqrt(2)*log(-sqrt(2)*e^(-x) + e^(-2*x) + 1) - 1/4*log(e^(
-x) + 1) + 1/4*log(e^(-x) - 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (18) = 36\).
time = 0.37, size = 54, normalized size = 2.08 \begin {gather*} \frac {1}{8} \, \sqrt {2} \log \left (\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} + 2 \, \sqrt {2} \cosh \left (x\right ) + 2}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2}}\right ) - \frac {1}{4} \, \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \frac {1}{4} \, \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*csch(4*x),x, algorithm="fricas")

[Out]

1/8*sqrt(2)*log((cosh(x)^2 + sinh(x)^2 + 2*sqrt(2)*cosh(x) + 2)/(cosh(x)^2 + sinh(x)^2)) - 1/4*log(cosh(x) + s
inh(x) + 1) + 1/4*log(cosh(x) + sinh(x) - 1)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \cosh {\left (x \right )} \operatorname {csch}{\left (4 x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*csch(4*x),x)

[Out]

Integral(cosh(x)*csch(4*x), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (18) = 36\).
time = 0.39, size = 57, normalized size = 2.19 \begin {gather*} -\frac {1}{8} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (-x\right )} - e^{x}}{\sqrt {2} + e^{\left (-x\right )} + e^{x}}\right ) - \frac {1}{8} \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) + \frac {1}{8} \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*csch(4*x),x, algorithm="giac")

[Out]

-1/8*sqrt(2)*log(-(sqrt(2) - e^(-x) - e^x)/(sqrt(2) + e^(-x) + e^x)) - 1/8*log(e^(-x) + e^x + 2) + 1/8*log(e^(
-x) + e^x - 2)

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Mupad [B]
time = 0.06, size = 61, normalized size = 2.35 \begin {gather*} \frac {\ln \left (\frac {1}{2}-\frac {{\mathrm {e}}^x}{2}\right )}{4}-\frac {\ln \left (-\frac {{\mathrm {e}}^x}{2}-\frac {1}{2}\right )}{4}+\frac {\sqrt {2}\,\ln \left (-\frac {{\mathrm {e}}^{2\,x}}{8}-\frac {\sqrt {2}\,{\mathrm {e}}^x}{8}-\frac {1}{8}\right )}{8}-\frac {\sqrt {2}\,\ln \left (\frac {\sqrt {2}\,{\mathrm {e}}^x}{8}-\frac {{\mathrm {e}}^{2\,x}}{8}-\frac {1}{8}\right )}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)/sinh(4*x),x)

[Out]

log(1/2 - exp(x)/2)/4 - log(- exp(x)/2 - 1/2)/4 + (2^(1/2)*log(- exp(2*x)/8 - (2^(1/2)*exp(x))/8 - 1/8))/8 - (
2^(1/2)*log((2^(1/2)*exp(x))/8 - exp(2*x)/8 - 1/8))/8

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